C-equivalence: Cross-Disciplinary Insights

• C-equivalence is a context-dependent term that defines distinct equivalence notions in electrodynamics, local relativity, singularity theory, and operator algebras.
• In classical electrodynamics, it links the static parameter c₍ᵤ₎ with the wave propagation speed c, ensuring consistency in Maxwell’s equations and Lorentz covariance.
• In singularity theory and operator algebras, it provides criteria for function germ equivalence and implements strong Morita equivalence via linking groupoids and Fell systems.

Searching arXiv for recent and foundational uses of “C-equivalence” across the relevant literatures. C-equivalence is a context-dependent term rather than a single cross-disciplinary concept. In classical electrodynamics it denotes the identification of the static quantity $c_u=1/\sqrt{\varepsilon_0\mu_0}$ with the observed propagation speed $c$ of electromagnetic disturbances; in a 2025 wave-mechanical formulation it denotes the compatibility of local pseudo-inertial charts by local Lorentz isometries; in singularity theory it appears in the broader notion of contact $(C)$-equivalence and in the specific relation of $C^r$-right equivalence of function germs; and in operator algebras it is used for strong Morita equivalence of reduced $C^*$-algebras associated with equivalent Fell systems (Heras, 2010, Kichenassamy, 8 Jul 2025, Migus, 2015, Moutuou et al., 2011).

1. Terminological scope and disciplinary uses

The expression has several technically distinct meanings, each tied to a different research program.

Domain	Meaning of “C-equivalence”	Representative source
Classical electrodynamics	$c_u=c$, where $c_u=1/\sqrt{\varepsilon_0\mu_0}$ is extracted from static laws and $c$ is the observed wave speed	(Heras, 2010)
Wave mechanics and local SR	A consistent atlas of pseudo-inertial charts whose overlap maps preserve the light cone and satisfy $(\phi_{ji})^*\eta=\eta$	(Kichenassamy, 8 Jul 2025)
Singularity theory	Contact $(C)$-equivalence in general; $c$0-right equivalence when only source diffeomorphisms are allowed	(Migus, 2015)
Operator algebras	Strong Morita equivalence of reduced $c$1-algebras arising from equivalent Fell systems	(Moutuou et al., 2011, Sims et al., 2011)

In the singularity-theoretic literature, the relevant letter is the $c$2 of differentiability or contact equivalence. In the electrodynamic literature, the relevant symbol is the speed $c$3. The shared notation therefore masks distinct objects: a universal velocity parameter, a Lorentz-compatible atlas, a germ-equivalence relation, and an imprimitivity relation between $c$4-algebras. This suggests that the term is best understood locally within each field rather than globally across mathematics and physics.

2. The $c$5-equivalence principle in classical electrodynamics

In SI units the vacuum permittivity $c$6 is defined by Coulomb’s law, and the vacuum permeability $c$7 by the Biot–Savart law for two long parallel currents. From these static, action-at-a-distance laws one constructs

$c$8

numerically $c$9. In the formulation under discussion, $(C)$0 is physically the speed that appears when consistency is demanded between electrostatic and magnetostatic force laws, and it has no direct dynamical or radiative meaning. By contrast, Maxwell’s dynamical field equations yield wave equations for $(C)$1 and $(C)$2 with a propagation constant $(C)$3, the observed speed of electromagnetic disturbances in vacuum. The statement

$(C)$4

is called the $(C)$5-equivalence principle (Heras, 2010).

The historical motivation given for this identification begins with Weber and Kohlrausch in 1856, who measured a ratio of electrostatic to electromagnetic units of charge and found $(C)$6. Kirchhoff noted the numerical coincidence with the speed of light, and Maxwell used the appearance of $(C)$7 in his wave equations to propose that light is an electromagnetic wave. The same source states that modern high-precision experiments, including cavity resonators, time-of-flight methods, and frequency comb methods, confirm $(C)$8 to better than $(C)$9 (Heras, 2010).

A central technical point is that Faraday’s law changes form if $C^r$0 is not assumed. Writing

$C^r$1

with $C^r$2 initially undetermined, and combining this with

$C^r$3

one obtains a $C^r$4-field wave equation. Requiring that its propagation speed be $C^r$5 fixes

$C^r$6

Accordingly, the cited paper writes the correct form of Faraday’s law without assuming $C^r$7-equivalence as

$C^r$8

With $C^r$9 and $C^*$0 kept distinct, the SI Maxwell equations are written as

$C^*$1

Under the identification $C^*$2, these reduce to the conventional SI form (Heras, 2010).

3. Covariance, gauge structure, and relation to the weak equivalence principle

A subsequent analysis argues that, in classical relativistic electrodynamics formulated in the mass–length–time system of units, there is only a single fundamental constant of nature, namely the invariant speed $C^*$3. The argument is based on the covariant action

$C^*$4

where the only free parameter introduced to fix the choice of electrical units is $C^*$5. In this formulation, $C^*$6 appears in the kinetic term and in the coupling of the four-potential to the four-current, and no other universal velocity may enter without spoiling the Minkowski structure of spacetime or gauge invariance. On this basis, the same work states that the principle $C^*$7 is not an extra postulate beyond classical relativistic electrodynamics plus special relativity, but rather a restatement of the uniqueness of the velocity parameter governing both electromagnetism and mechanics (Choy, 2011).

The relation to Einstein’s second postulate is made explicit through the gauge transformations

$C^*$8

These are said to be consistent with the four-vector structure $C^*$9 only if no other speed scale $c_u=c$0 enters. Replacing $c_u=c$1 by $c_u=c$2 in the gauge law for $c_u=c$3 would, in the cited argument, destroy the usual Lorentz covariance of Maxwell’s equations (Choy, 2011).

The same paper considers an alternative electrodynamics with $c_u=c$4, leading to a particle Lagrangian

$c_u=c$5

Its conclusion is that, unless $c_u=c$6, the scaling of physical laws under arbitrary changes of units is violated. A further step couples this extended toy model to Newtonian gravity and imposes both invariance under overall unit rescalings and equality of inertial and gravitational mass, $c_u=c$7. Within that context, the authors argue that the weak equivalence principle and the $c_u=c$8-equivalence principle are two facets of the same scaling symmetry. The claim is therefore not merely empirical equality but structural necessity within the cited framework (Choy, 2011).

This position differs in emphasis from the earlier electrodynamic presentation. The earlier work treats $c_u=c$9 as an experimentally established principle whose omission changes the form of Faraday’s law; the later work argues that a consistent relativistic electrodynamics in MLT units admits only one fundamental speed in the first place. The contrast is a genuine point of interpretation within the literature (Heras, 2010, Choy, 2011).

4. C-equivalence in wave mechanics and local special relativity

A 2025 formulation introduces C-equivalence in a different sense. Each physical observer $c_u=1/\sqrt{\varepsilon_0\mu_0}$0 is assigned a local chart

$c_u=1/\sqrt{\varepsilon_0\mu_0}$1

where $c_u=1/\sqrt{\varepsilon_0\mu_0}$2 carries the flat Lorentzian metric

$c_u=1/\sqrt{\varepsilon_0\mu_0}$3

Such a chart is called a C-system or pseudo-inertial system. Whenever two observers can exchange light signals so that $c_u=1/\sqrt{\varepsilon_0\mu_0}$4, the overlap map

$c_u=1/\sqrt{\varepsilon_0\mu_0}$5

must preserve the light-cone structure, equivalently

$c_u=1/\sqrt{\varepsilon_0\mu_0}$6

The collection $c_u=1/\sqrt{\varepsilon_0\mu_0}$7 is then said to form a consistent C-equivalent atlas. In the smooth category, this forces each overlap to be a local Lorentz isometry of $c_u=1/\sqrt{\varepsilon_0\mu_0}$8, possibly depending on position in the overlap region (Kichenassamy, 8 Jul 2025).

Within this framework, de Broglie’s wave is interpreted as furnishing its own local space-time units. The phase wave

$c_u=1/\sqrt{\varepsilon_0\mu_0}$9

attaches to each event an internal proper-time parameter $c$0. Because phase is a Lorentz scalar, in any local chart of the surrounding Minkowski patch one has

$c$1

and therefore

$c$2

The paper interprets this as showing that the oscillation frequency $c$3 determines a one-parameter foliation $c$4 and, through the spatial wave-number $c$5, an internal Euclidean metric on each $c$6-slice (Kichenassamy, 8 Jul 2025).

The cited treatment emphasizes invariance relations. In one spatial dimension,

$c$7

and under a Lorentz boost one obtains transformed quantities $c$8 and $c$9 such that $(\phi_{ji})^*\eta=\eta$0. The same source states that the proper frequency $(\phi_{ji})^*\eta=\eta$1 is the same in all C-equivalent charts. Using the Einstein–de Broglie relations

$(\phi_{ji})^*\eta=\eta$2

it derives the group velocity

$(\phi_{ji})^*\eta=\eta$3

so that the wave packet’s built-in velocity coincides with the tangent of its C-system world-line (Kichenassamy, 8 Jul 2025).

The same work extends the construction to accelerated motion. For a world-line $(\phi_{ji})^*\eta=\eta$4 with

$(\phi_{ji})^*\eta=\eta$5

one erects a local orthonormal tetrad satisfying Fermi–Walker transport,

$(\phi_{ji})^*\eta=\eta$6

In instantaneous coordinates $(\phi_{ji})^*\eta=\eta$7, the metric takes the Minkowski form up to first order in acceleration: $(\phi_{ji})^*\eta=\eta$8 Illustrative cases listed in the paper include uniform translation, non-uniform acceleration described through Frenet–Serret formulas of four-dimensional acceleration, optical resonance in an extended atom, and electron–molecule “single-electron” interferences. The interpretation offered there is that one must distinguish the variety of events from the variety of observers, and that overlaps generated by light exchange knit local Minkowski patches into global structure (Kichenassamy, 8 Jul 2025).

5. $(\phi_{ji})^*\eta=\eta$9-right equivalence and contact $(C)$0-equivalence in singularity theory

In singularity theory, the relevant usage is not the speed $(C)$1 but the $(C)$2 differentiability class and the broader notion of contact $(C)$3-equivalence. Let $(C)$4 denote the ring of $(C)$5-germs

$(C)$6

and let $(C)$7 be the Jacobi ideal generated by $(C)$8. Two map-germs

$(C)$9

are called $c$00-right equivalent if there exists a $c$01-diffeomorphism

$c$02

such that

$c$03

in some neighborhood of $c$04. The cited paper notes that the most general notion in this setting is contact $c$05-equivalence, where one allows source-coordinate changes together with target-coordinate changes; its theorem restricts to right equivalence, namely $c$06 changes in the source only (Migus, 2015).

The main theorem concerns $c$07-germs $c$08 and $c$09 with $c$10 and $c$11. If there exist a neighborhood $c$12 of $c$13 and a constant $c$14 such that for every multi-index $c$15 with $c$16,

$c$17

then there exists a local $c$18-diffeomorphism $c$19 such that

$c$20

for $c$21 near $c$22. The result provides a concrete sufficient criterion for deciding when two germs lie in the same $c$23-right equivalence class (Migus, 2015).

The proof follows the Kuiper–Kuo method of integrating a suitable vector field. One introduces the homotopy

$c$24

derives comparison estimates

$c$25

defines

$c$26

and solves the ODE

$c$27

Along its solutions $c$28, one has

$c$29

hence $c$30, or $c$31. Auxiliary ingredients include a Łojasiewicz-type estimate, derivative estimates for $c$32, and a uniqueness-of-solutions lemma across the singular set $c$33 (Migus, 2015).

The paper also gives an algebraic criterion: if

$c$34

then the gradient inequality above holds automatically, so $c$35 and $c$36 are $c$37-right equivalent. For

$c$38

any perturbation

$c$39

with $c$40 homogeneous of degree at least $c$41, satisfies the criterion. A separate $c$42-case is obtained under local Lipschitz hypotheses on $c$43 and $c$44 together with bounds on $c$45 and $c$46 (Migus, 2015).

6. C-equivalence as strong Morita equivalence for Fell systems and reduced $c$47-algebras

In the operator-algebraic literature, “C-equivalence” is explicitly identified with strong Morita equivalence in the sense of Rieffel for reduced $c$48-algebras arising from equivalent Fell systems (Moutuou et al., 2011). A Fell system over a groupoid $c$49 consists of a continuous Banach bundle $c$50 with fibrewise multiplication and involution satisfying the usual norm, associativity, positivity, and fullness conditions. Its convolution $c$51-algebra $c$52 is completed in the reduced norm coming from the left-regular representation on the Hilbert $c$53-module $c$54, where

$c$55

to obtain

$c$56

If $c$57 and $c$58 are Morita equivalent groupoids and $c$59, $c$60 are Fell bundles, a Fell pair over an $c$61-bibundle $c$62 consists of a Banach bundle $c$63, compatible left and right actions, and $c$64-valued and $c$65-valued inner products making each fibre $c$66 an imprimitivity bimodule. One then writes

$c$67

The reduced equivalence theorem states that from such data one forms the linking groupoid

$c$68

and the linking Fell bundle

$c$69

for which $c$70 contains complementary full projections $c$71 and $c$72 satisfying $c$73. The corner algebras are naturally isomorphic to the original reduced $c$74-algebras,

$c$75

and the off-diagonal corner

$c$76

is an imprimitivity bimodule implementing

$c$77

This is the operator-algebraic content of C-equivalence in that paper (Moutuou et al., 2011).

The corresponding dense bimodule is $c$78, with convolution-type actions and inner products. Completing it in the $c$79-norm

$c$80

yields a full imprimitivity bimodule $c$81. The result is summarized there as the statement that the reduced $c$82-functor sends Morita equivalences of Fell systems to strong Morita equivalences of reduced $c$83-algebras (Moutuou et al., 2011).

A parallel theorem for upper semicontinuous Fell bundles over groupoids constructs a linking bundle $c$84 over a linking groupoid $c$85 from an equivalence $c$86 of Fell bundles $c$87 and $c$88. Its full and reduced cross-sectional algebras contain the original algebras as complementary full corners, and the universal-to-reduced quotient on the linking algebra restricts on each corner to the relevant reduced quotient. In particular, the full symmetric imprimitivity theorem passes through to reduced crossed products, generalizing the Quigg–Spielberg result to Fell bundles over groupoids (Sims et al., 2011).

Taken together, these results fix the operator-algebraic meaning of C-equivalence as an imprimitivity relation implemented by linking groupoids, linking Fell bundles, and corner realizations inside reduced linking algebras. Here the term has no relation to the velocity $c$89 of electrodynamics or to $c$90-equivalence of smooth germs; it names a precise Morita-theoretic equivalence class of reduced $c$91-algebras (Moutuou et al., 2011, Sims et al., 2011).